The purpose of miracles is to confirm that Jesus of Nazareth was the Messiah of Israel
Act 4:1-22 NASB - "1 As they were speaking to the people, the priests and the captain of the temple guard and the Sadducees came up to them, 2 being greatly disturbed because they were teaching the people and proclaiming in Jesus the resurrection from the dead. 3 And they laid hands on them and put them in jail until the next day, for it was already evening.
4 But many of those who had heard the message believed; and the number of the men came to be about five thousand. 5 On the next day, their rulers and elders and scribes were gathered together in Jerusalem; 6 and Annas the high priest was there, and Caiaphas and John and Alexander, and all who were of high-priestly descent.
7 When they had placed them in the center, they began to inquire, "By what power, or in what name, have you done this?"
8 Then Peter, filled with the Holy Spirit, said to them, "Rulers and elders of the people, 9 if we are on trial today for a benefit done to a sick man, as to how this man has been made well, 10 let it be known to all of you and to all the people of Israel, that by the name of Jesus the Messiah, the Nazarene, whom you crucified, whom God raised from the dead--by this name this man stands here before you in good health. 11 "He is the STONE WHICH WAS REJECTED by you, THE BUILDERS, but WHICH BECAME THE CHIEF CORNER stone. 12 "And there is salvation in no one else; for there is no other name under heaven that has been given among men by which we must be saved."
13 Now as they observed the confidence of Peter and John and understood that they were uneducated and untrained men, they were amazed, and began to recognize them as having been with Jesus.
14 And seeing the man who had been healed standing with them, they had nothing to say in reply. 15 But when they had ordered them to leave the Council, they began to confer with one another, 16 saying, "What shall we do with these men? For the fact that a noteworthy miracle has taken place through them is apparent to all who live in Jerusalem, and we cannot deny it.
17 "But so that it will not spread any further among the people, let us warn them to speak no longer to any man in this name." 18 And when they had summoned them, they commanded them not to speak or teach at all in the name of Jesus.
19 But Peter and John answered and said to them, "Whether it is right in the sight of God to give heed to you rather than to God, you be the judge; 20 for we cannot stop speaking about what we have seen and heard."
21 When they had threatened them further, they let them go (finding no basis on which to punish them) on account of the people, because they were all glorifying God for what had happened; 22 for the man was more than forty years old on whom this miracle of healing had been performed."
1Corinthians 1:22-24 NASB - "22 For indeed Jews ask for signs and Greeks search for wisdom; 23 but we preach Messiah crucified, to Jews a stumbling block and to Gentiles foolishness, 24 but to those who are the called, both Jews and Greeks, Messiah the power of God and the wisdom of God."
Musing & Pondering
First, we must understand that Luke places this story of a healing of a lame man from birth in real time and space history and not as a myth intended to teach some abstract moral or philosophical truth.
Josephus the Jewish historical talks about Annas and Caiaphas related to being High Priests.  So it is good for us to mention that Luke is speaking about reality and this is not some “salvation history” which has nothing to do with normal history and that if they had video cameras the could have recorded these events and conversations.
The problem they have here is that they have found Jesus of Nazareth to be a false prophet and Messiah and therefore have encouraged the Romans to crucify HIM as a threat to Creaser’s rule. If this is true then the “name” which means “using the authority of” would be powerless to heal anyone. False prophets and Messiah’s have no spiritual authority or power once they are dead. So how did Peter and John us the authority of a dead false Messiah to heal a man lame since birth?
The only answer that seems to make sense is that Jesus of Nazareth was not a false Messiah but the real anointed prophet, priest, and king of God. The horror of that is that this would mean they have rejected God’s chosen King and Savior. In addition then this further confirms the eyewitness reports that God raised Jesus from the dead because here is the living power and authority of Messiah Jesus demonstrated by the healing of this man. 
Peter quotes Psalm 118:22 here based apparently on a Jewish understanding that at the time of Solomon the builders of the first temple had at first rejected the stone carved out to be the cornerstone of the temple. Peter sees in this a parallel to how the Messiah Jesus who is the cornerstone of God’s new spiritual temple was also rejected by those who were to building the true faith instead of opposing it.
Peter makes a bold claim that salvation can only come by the authority of the crucified and resurrected Messiah Jesus of Nazareth. Deliverance from the curse of humanities rebellion against the kingdom of God is exclusively offered in the acceptance of Messiah Jesus. Why would or could that be.
Well salvation is only needed where there is people are in need of deliverance, forgiveness, and restoration. If all is well no salvation is needed. There are three ways we can hope to be saved from a bad situation.
One we can hope that we can pull ourselves up by our own bootstraps and by hard work, focus, and effort save ourselves from the bad situation we are in and avoid then the oppression, condemnation, and destruction that we are in danger of or even experiencing. This would be “Self Salvation” or “Legalistic Salvation”. No savior is needed but perhaps a life coach might help. Most every religions created by human beings fits under this approach to the issue of salvation.
The other it that someone will save us totally not related to our deserving salvation. Some people hope that luck, good fortune, or fate may save them. Others put their hope in a generic God that will somehow take away the just consequences of their immoral actions. But in such a case God is merciful but not just. This is “Lawless Salvation” and feels that justice will have to be forgotten if we are to have any hope of avoiding horrible consequences
The final solution to this problem is found only in the teaching of the Apostles. The answer is that God is so dedicated to justice that the curse must be paid by the Messiah taking the place of HIS people. The full demands of justice have to be met and are met in the suffering and death of the Messiah Jesus. But this sacrifice now allows God to justly give salvation by grace to all who find refuge and hope in the Messiah by accepting HIM and HIS sacrifice in their place. This is the message of the Christian faith in Messiah Jesus.
Now God will provide justice for all (Read Romans 2). Those with the Bible and those without the Bible will be given exactly the reward or punishment that their deeds deserve. But the reality is there is none who are good morally. All humanity has fallen short of following their conscience and being honest with the revelation they have been given. So getting justice while fair is not very hopeful in providing salvation. That is why a saving Messiah was needed because we had all utterly failed. For this reason the only hope of salvation is taking refuge and trusting in Jesus only Jesus to save us.
So this it totally fair and yet provides hope for all nations. It does mean we need to try to convert all people to the truth of Christianity and abandon all hope in every other means of salvation. For the Messiah Jesus is the only way.
This message was so radical that the Jewish authorities attempted by intimidation to silence it. They did not mind if miraculous miracles occurred just as long as they did were not done in the authority of the Messiah Jesus. Proof of the Apostles conviction concerning Jesus is that threats did not silence them.
How should we then live?
First, it is clear that we should seriously consider the claims of the Messiah Jesus and if we find them reasonable, personally accept HIM as our prophet, priest, and king. We should be converted to the Christina faith.
Second it is clear that we should pray that the Lord will give us opportunity to share with others the truth about Messiah Jesus that they may also consider the truth of HIS claims and after verifying them to be valid, accepting HIM as their Messiah.
Third, we need to be prepared and expecting persecution due to this message. We should not act rude and rough hoping to inspire persecution. If we are acting like religious jerks then our being mistreated is not for Jesus but is simply the result of acting in a uncivil and unloving manner. You don’t get rewards for suffering the social consequences of being rude and rough with others.. But if we speak the truth in love then we must still be ready to suffer abuse by those who disagree. This is normal because our message attacks all other hopes of salvation.
2Ti 2:24-26 NASB - "24 The Lord's bond-servant must not be quarrelsome, but be kind to all, able to teach, patient when wronged, 25 with gentleness correcting those who are in opposition, if perhaps God may grant them repentance leading to the knowledge of the truth, 26 and they may come to their senses and escape from the snare of the devil, having been held captive by him to do his will."
Lord, help us to be ready to be persecuted for your sake and the sake of the gospel. Help us show respect, patience, and love to those who reject the gospel. Grant to us a true faith in YOU and YOUR sacrifice for our sins. Help us to be bold in our witness and demonstrate our living authority and power through us. Allow our witness to be used to create true faith in others. Amen
1.  Josephus' source is mentioned in Antiquitates Judaicae 20.224-51 and Against Apion 1.36; see Bond, Caiaphas, p. 163, n. 2.
 See my blog on the problem of miracles for those grieving. http://musingsonmourning.blogspot.com/
Also C.S. Lewis book on Miracles and Miracles: What They Are, Why They Happen, and How They Can Change Your Life by Eric Metaxas